To solve the problem, we will follow these steps:
### Step 1: Identify the Circuit Configuration
We have two resistors, \( R_1 = 100 \, \Omega \) and \( R_2 = 200 \, \Omega \), connected in series with a voltage source of \( V = 220 \, V \). A voltmeter with a resistance of \( R_v = 1000 \, \Omega \) is connected in parallel to the \( 100 \, \Omega \) resistor.
### Step 2: Calculate the Equivalent Resistance of the Parallel Combination
The voltmeter and the \( 100 \, \Omega \) resistor are in parallel. The formula for the equivalent resistance \( R_{eq} \) of two resistors in parallel is given by:
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_v}
\]
Substituting the values:
\[
\frac{1}{R_{eq}} = \frac{1}{100} + \frac{1}{1000}
\]
Finding a common denominator (1000):
\[
\frac{1}{R_{eq}} = \frac{10}{1000} + \frac{1}{1000} = \frac{11}{1000}
\]
Now, taking the reciprocal to find \( R_{eq} \):
\[
R_{eq} = \frac{1000}{11} \, \Omega
\]
### Step 3: Calculate the Total Resistance in the Circuit
The total resistance \( R_{total} \) in the circuit is the sum of the equivalent resistance \( R_{eq} \) and the \( 200 \, \Omega \) resistor:
\[
R_{total} = R_{eq} + R_2 = \frac{1000}{11} + 200
\]
To add these, convert \( 200 \) into a fraction with a denominator of 11:
\[
200 = \frac{2200}{11}
\]
Now adding:
\[
R_{total} = \frac{1000}{11} + \frac{2200}{11} = \frac{3200}{11} \, \Omega
\]
### Step 4: Calculate the Current in the Circuit
Using Ohm's law, the current \( I \) flowing through the circuit can be calculated as:
\[
I = \frac{V}{R_{total}} = \frac{220}{\frac{3200}{11}} = 220 \times \frac{11}{3200}
\]
Calculating this gives:
\[
I = \frac{2420}{3200} = \frac{121}{160} \, A
\]
### Step 5: Calculate the Voltage Across the \( 100 \, \Omega \) Resistor
The voltage \( V_{100} \) across the \( 100 \, \Omega \) resistor (which is also the reading on the voltmeter) can be calculated using Ohm's law:
\[
V_{100} = I \times R_{eq} = \left(\frac{121}{160}\right) \times \left(\frac{1000}{11}\right)
\]
Calculating this gives:
\[
V_{100} = \frac{121 \times 1000}{160 \times 11} = \frac{121000}{1760}
\]
Now simplifying:
\[
V_{100} = 68.75 \, V
\]
### Final Answer
The reading of the voltmeter is \( 68.75 \, V \).
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